2. A divisor of one of two numbers, and not of the other, will divide neither their sum nor their difference.

For, one number will be a whole number of times the divisor, and the other a mixed number of times the divisor, and consequently neither their sum nor their difference will be a whole number of times the divisor, since neither the sum nor the difference of an integer and mixed number can be an integer.

3. A number which is not a divisor of either of two numbers may or may not divide their sum or their difference.

Any two such numbers will equal a number of times the assumed number, plus certain remainders. Now, if the sum of these remainders equals the number, the sum of the numbers is evidently divisible by the number; and if the difference of these remainders is zero, the difference of the numbers will be divisible by the number. In all other cases, neither the sum nor difference of the numbers is divisible by the number.

COMPOSITE NUMBERS.

955. A Composite Number is one which can be produced by multiplying together two or more numbers, each of which is greater than a unit.

956. The Principles of composite numbers are the truths which state their relation to their factors. These principles enable us to determine their factors or divisors.

PRINCIPLES.

1. A number is divisible by 2 when the right hand term is zero or an even digit.

If the right hand digit is zero, the number equals a number of tens; and, since 10 is divisible by 2, any number of tens is divisible by 2.

Any number may be separated into two parts a multiple of ten plus the right hand digit—and when the right hand digit is divisible by 2, both of these parts are divisible by 2, hence their sum, which is the number itself, is divisible by 2 (Prin. 1, Art. 954).

2. A number is divisible by 3 when the sum of its digits is divisible by 3.

In Prin. 8, it will be shown that every number consists of a multiple of 9, plus the sum of its digits; hence since a multiple of 9 is divisible by 3, when the sum of the digits is divisible by 3, the number itself is divisible by 3.

3. A number is divisible by 4 when the two right hand terms are ciphers, or when they express a number which is divisible by 4.

If the two right hand terms are ciphers, the number equals a number

of hundreds, and since 100 is divisible by 4, any number of hundreds is divisible by 4.

Any number may be separated into two parts- a number of hundreds, plus the number expressed by the two right hand digits (thus 1232=1200+ 32); and when the number expressed by the two right hand digits is divisible by 4, both of the parts are divisible by 4, hence their sum, which is the number itself, is divisible by 4 (Prin. 1, Art. 954).

4. A number is divisible by 5 when its right hand term is 0 or 5.

If the right hand term is 0, the number is a number of times 10, and since 10 is divisible by 5, the number itself is divisible by 5.

If the right hand term is 5, the entire number will consist of a number of tens plus 5, and since both of these are divisible by 5, their sum, which is the number itself, is divisible by 5.

5. A number is divisible by 6 when it is even, and the sum of the digits is divisible by 3.

Since the number is even it is divisible by 2, and since the sum of the digits is divisible by 3, the number is divisible by 3, and since it contains both 2 and 3, it will contain their product 3×2, or 6 (Prin. 3, Art. 165).

6. A number is divisible by 7 when the sum of the odd numerical periods minus the sum of the even numerical periods is divisible by 7.

Take any number, as 7936367225. This can be resolved, as shown below, into a multiple of 7, plus the difference between the sums of the odd numerical periods and the even numerical periods. For 1001 is a multiple of 7, 999999 is 999 times 1001, 1000000001 is also a multiple of 1001, and carrying out the number to higher periods, we shall continue to have multiples of 1001, alternately 1 more and 1 less than the number represented by the unit of the period. In the same way it may be shown that any number is equal to a multiple of 7 plus the difference between the odd and even numerical periods; hence when the difference between those periods is divisible by 7, the number is divisible by 7.

7936367225=

225

+225 1001-367

367000=367 × (1001-1)=367X 936000000=936× (999999+1)=936×999999+936 7000000000=7× (1000000001 -1)=7× 1000000001 -7 7×1000000001+936× 999999+367×1001–7+936-367+225 7. A number is divisible by 8 when the three right hand terms are ciphers, or when the number expressed by them is divisible by 8.

If the three right hand terms are ciphers, the number equals a number of thousands, and since 1000 is divisible by 8, any number of thousands is divisible by 8.

A number may be resolved into a number of thousands plus the number expressed by the three right hand digits (thus 17368=17000+368); and when both of these parts are divisible by 8, their sum, which is the number itself, is divisible by 8.

8. A number is divisible by 9 when the sum of the digits

is divisible by 9.

Take any number, as 567. This can be resolved, as shown.in the margin, into (5× 9969)+(5+ 6 +7), the first part of

7

567= 60 6× 10=6×( 9+1)=6× 9+6

500 = 567=

=5X100

5X (99+1)=5×99+5 599+6x9+5+6+7

In

which is divisible by 9, and the other part is the sum of the digits. the same way it may be shown that any number is equal to a multiple of 9 plus the sum of the digits; hence, when the sum of the digits is divisible by 9, the number is divisible by 9.

9. A number is divisible by 10 when the unit figure is 0. For, such a number equals a number of tens, and any number of tens is divisible by 10, hence the number is divisible by 10.

10. Any number is divisible by 11, when the difference between the sums of the digits in the odd places and in the even places is divisible by 11, or when this difference is 0.

of the digits in the odd places and the even places. In the same way it may be shown that any number equals a multiple of 11, plus the difference between the sums of the digits in the odd and even places; hence, when the difference between these sums is divisible by 11, or is 0, the number is divisible by 11.

11. A number is divisible by 12 when the sum of the digits is divisible by 3 and the number expressed by the two right hand digits is divisible by 4.

For, since the sum of the digits is divisible by 3, the number is divisible by 3, and since the number expressed by the two right hand digits is divisible by 4, the number is divisible by 4; hence, since the number is divisible by both 3 and 4, it is divisible by their product, or 12.

NOTE. In a similar manner we can find conditions of divisibility by 14, 15, 16, 18, etc. It will be an interesting exercise for the pupils to state such conditions. The subject, however, is more theoretical than practical.

EXAMPLES FOR PRACTICE.

Name some of the divisors of the following numbers:

PRIME NUMBERS.

957. A Prime Number is a number that cannot be produced by multiplying two or more numbers together, each of which is greater than a unit.

958. The Principles of prime numbers are the truths which enable us to determine primes.

PRINCIPLES.

1. A prime number has no integral divisor except itself and unity.

For, if it had, it would be the product of two numbers, each greater than unity, and hence would not be a prime number.

2. Every prime number except 2 is an odd number. For, if it is not odd it is even, but if even it is divisible by 2, and hence not prime; therefore any prime number except 2 must be odd.

3. The right hand term of every prime number, except 2 and 5, must be 1, 3, 7, or 9.

For, if the right hand term is even, the number is divisible by 2, and if it is 5 or 0, it is divisible by 5, in both of which cases it is not prime. 4. If a number has no integral divisor not exceeding its square root, it is a prime number.

For, if a number has no divisor less than its square root, it cannot have one greater than its square root, since if it had, the quotient would be a divisor, and it would thus have a divisor less than its square root.

5. Every prime number greater than 2 is a multiple of 4, plus 1, or minus 1.

For, if we divide a prime number by 4, the remainder may be either 1, 2, or 3; hence a prime number equals a number of times 4,+1, or +2, or +3. But a number of times 4, +2 is divisible by 2, and hence is not prime; therefore every prime number must be a number of times 4, +1, or a number of times 4,+3. But a number of times 4,+3 is also a number of times 4,-1.

6. Every prime number greater than 3 is a multiple of 6, plus or minus 1.

If we divide a prime number by 6, the remainder must be either 1, 2, 3, 4, or 5; but the remainder cannot be 2, 3, or 4, for then the prime number would equal a number of times 6,+2, or a number of times 6,+3, or a number of times 6,+4, the second of which is divisible by 3, and the others by 2; hence the remainder must be 1 or 5, and consequently every prime number equals a number of times 6,+1, or a number of times 6,—1.

7. Every prime number greater than 5 is a multiple of 8, plus 1 or 3, or minus 1 or 3.

The demonstration is similar to that of Prin. 6. prove it.

Let the pupil

NOTE.-Every prime number is comprehended in one or another of the above propositions, although the converse proposition, that every number in one of those forms is prime, is not true.

EXAMPLES FOR PRACTICE.

Show that these principles are true with the following primes:

959. A General Method of determining prime numbers beyond a certain limit, has not yet been discovered, although much time has been spent in the investigation.

960. The method commonly used consists in writing a series of numbers and sifting out those which are composite, the remaining numbers being prime.

CASE I.

961. To find all the prime numbers from 1 up to any limit.

1. Find the prime numbers below 100.

METHOD. Since all the prime numbers except 2 are odd (Prin. 2.), we write the series of odd numbers thus:

3

3

3.5

3.7

5 3

3 5.7

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37,

3.5

7 3

5 3

3.7 5

3

39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73,

75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99.

Now, since this series increases by 2, the third term from 3 is 3+3×2, which is divisible by 3, hence every third term after 3 is divisible by 3, and is therefore composite. We will therefore place the figure 3 over every third term. We see by a similar course of reasoning, that every fifth term after 5 is divisible by 5, and is therefore composite; and will therefore place the figure 5 over every 5th number. Proceeding in the same manner with 7, the numbers unmarked, together with the number 2, will be the prime numbers below 100. Hence all the prime numbers below 100 are 1, 2, 3, 5, 7, 11, 13, etc., to 97.

NOTE. This method of finding prime numbers originated with Eratosthenes, a Greek mathematician. He inscribed the series of odd numbers upon parchment, and then cut out the composite numbers, leaving the primes. The parchment with its holes resembled a sieve; hence the method was called Eratosthenes' sieve.